MOTION OF A PARTICLE 291 Case I. The orbit is a parabola, when v<? = — ' r0 2 k Case II. The orbit is an ellipse, when v<?< — TQ 2 k Case III. The orbit is a hyperbola, when v<?> -- r° The general expression for the velocity, which is given by equation (9), may be put in the following special forms: 2k I. v* = — ^ when the orbit is a parabola. (2 ]\ --- ), when the orbit is an ellipse. r ar (2 1\ - + -), when the orbit is a hyperbola. / a/ The quantity a is the length of the semi-transverse axis. 222. Velocity from Infinity. — The velocity which the particle acquires in falling towards the center from a point infinitely distant from the center is called the velocity from infinity. This velocity may be computed from the energy equation. Thus ^mv2= I F dr J<X3 r m , -J'f* T Therefore v*=—- r But the last equation is identical with the relation which gives the velocity of a particle moving in a parabolic path, therefore if a particle describes a parabolic orbit its velocity at any point of its orbit is equal to the velocity it would have acquired if it had started from infinity and arrived at that